# Cyclotron | particle energy

## Particle energy

Since the particles are accelerated by the voltage many times, the final energy of the particles is not dependent on the accelerating voltage but on the strength of the magnetic field and the diameter of the accelerating chamber, the dees. Cyclotrons can only accelerate particles to speeds much slower than the speed of light, nonrelativistic speeds. For nonrelativistic particles, the centripetal force ${\displaystyle F_{C}\;}$ required to keep them in their curved path is

${\displaystyle F_{C}={mv^{2} \over r}\;}$

where ${\displaystyle m\;}$ is the particle's mass, ${\displaystyle v\;}$ its velocity, and ${\displaystyle r\,}$ is the radius of the path. This force is provided by the Lorentz force ${\displaystyle F_{B}\,}$ of the magnetic field ${\displaystyle B\,}$

${\displaystyle F_{B}=qvB\;}$

where ${\displaystyle q\,}$ is the particle's charge. The particles reach their maximum energy at the periphery of the dees, where the radius of their path is ${\displaystyle r\;=\;R}$ the radius of the dees. Equating these two forces

${\displaystyle {mv^{2} \over R}=qvB\;}$
${\displaystyle v={qBR \over m}\,}$

So the output energy of the particles is

${\displaystyle E={1 \over 2}mv^{2}={\frac {q^{2}B^{2}R^{2}}{2m}}\;}$

Therefore, the limit to the cyclotron's output energy for a given type of particle is the strength of the magnetic field ${\displaystyle B}$, which is limited to about 2 T for ferromagnetic electromagnets, and the radius of the dees ${\displaystyle R}$, which is determined by the diameter of the magnet's pole pieces. So very large magnets were constructed for cyclotrons, culminating in Lawrence's 1946 synchrocyclotron, which had pole pieces 4.67 m (184 in) (15.3 feet) in diameter.

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